arXiv subjects
Publications and source records attributed to James A. Mingo.
We elucidate a bijection between ribbon graphs on the real projective plane and non-crossing annular pairings that relate to the $1/N$ correction term of the GOE and LOE spectral moments. We also derive analogous objects for the $1/N^2$ correction terms of said moments and their equivalents for the GUE and LUE.
We compute the fluctuation moments $α_{m_1,\dots,m_r}$ of a Complex Wigner Matrix $X_N$ given by the limit $\lim_{N\rightarrow\infty}N^{r-2}k_r(Tr(X_N^{m_1}),\dots,Tr(X_N^{m_r}))$. We prove the limit exists and characterize the leading order via planar graphs that result to be trees. We prove these graphs can be counted by the set of non-crossing partitioned permutations which permit us to express the moments $α_{m_1,\dots,m_r}$ in terms of simpler quantities $κ_{m_1,\dots,m_r}$ known as the higher order cumulants. As for lower order dimensions ($r \leq 3$) we observe that while the moments have a more elaborated expression the cumulants are simpler.
We compute the limit distribution of partial transposes (when both the number and the size of blocks tends to infinity) for a large class of ensembles of unitarily invariant random matrices. Furthermore, it is shown the asymptotic freeness relation between the ensembles of random matrices, their transposes and their left and right partial transposes.
In an infinitesimal probability space we consider operators which are infinitesimally free and one of which is infinitesimal, in that all its moments vanish. Many previously analysed random matrix models are captured by this framework. We show that there is a simple way of finding non-commutative distributions involving infinitesimal operators and apply this to the commutator and anticommutator. We show the joint infinitesimal distribution of an operator and an infinitesimal idempotent gives us the Boolean cumulants of the given operator. We also show that Boolean cumulants can be expressed as infinitesimal moments thus giving matrix models which exhibit asymptotic Boolean independence and monotone independence. Finally we demonstrate a connection to the Markov-Krein transform.
We introduce $R$-diagonal and even operators of second order. We give a formula for the second order free cumulants of the square $x^2$ of a second order even element in terms of the second order free cumulants of $x$. Similar formulas are proved for the second order free cumulants of $aa^*$, when $a$ is a second order $R$-diagonal operator. We also show that if $r$ is second order $R$-diagonal and $b$ is second order free from $r$, then $rb$ is also second order $R$-diagonal. We present a large number of examples, in particular the limit distribution of products of Ginibre matrices. We prove the conjectured formula of Dartois and Forrester for the fluctuations moments of the product of two independent complex Wishart matrices and generalize it to any number of factors.
We compute the third order moments of a complex Wigner matrix. We provide a formula for the third order moments $α_{m_1,m_2,m_3}$ in terms of quotient graphs $T_{m_1,m_2,m_3}^π$ where $π$ is the Kreweras complement of a non-crossing pairing on the annulus. We prove that these graphs can be counted using the set of partitioned permutations, this permits us to write the third order moments in terms of the high order free cumulants which have a simple expression.
Let $X_N$ be a $N \times N$ real Wishart random matrix with aspect ratio $M/N$. The limit eigenvalue distribution of $X_N$ is the Marchenko-Pastur law with parameter $c = \lim_N M/N$. The limit moments $\{m_n\}_n$ are given by $m_n = \sum_π c^{\#(π)}$ where the sum runs over $NC(n)$. Let $m_n'$ be the limit of $N( \mathrm{E}(\mathrm {tr}(X_N^n)) - m_n)$. These are the asymptotic infinitesimal moments of a real Wishart matrix. We show that $m'_n$ can be written as a sum over planar diagrams with two terms, $\sum_π c'(\#(π) -1) c^{\#(π)-1}$, and $\sum_{π\in S_{NC}^δ(n,-n)} c^{\#(π)/2}$, where $S_{NC}^δ(n,-n)$ is a set of non-crossing annular permutations satisfying a symmetry condition. Moreover we present a recursion formula for the second term which is related to one for higher order freeness.
In this paper we propose a new approach to the central limit theorem (CLT), based on functions of bounded Féchet variation for the continuously differentiable linear statistics of random matrix ensembles which relies on: a weaker form of a large deviation principle for the operator norm; a Poincaré-type inequality for the linear statistics; and the existence of a second-order limit distribution. This approach frames into a single setting many known random matrix ensembles and, as a consequence, classical central limit theorems for linear statistics are recovered and new ones are established, e.g., the CLT for the continuously differentiable linear statistics of block Gaussian matrices. In addition, our main results contribute to the understanding of the analytical structure of second-order non-commutative probability spaces. On the one hand, they pinpoint the source of the unbounded nature of the bilinear functional associated to these spaces; on the other hand, they lead to a general archetype for the integral representation of the second-order Cauchy transform, $G_2$. Furthermore, we establish that the covariance of resolvents converges to this transform and that the limiting covariance of analytic linear statistics can be expressed as a contour integral in $G_2$.
In this article, we shall start with a closed walk on a regular tree of degree $d$. These walks are described by the Kesten-McKay law which arises as the asymptotic distribution of a random $d$-regular graph on $n$ vertices. We will show that the moments of the Kesten-McKay law are given by counting standard Young tableaux with at most 2 rows, and how some properties of the walk make sense even when $d$ is not an integer. We will use free probability to instruct us how to build an explicit model in random matrix theory.
We consider the effect of a partial transpose on the limit $*$-distribution of a Haar distributed random unitary matrix. If we fix, $b$, the number of blocks, we show that the partial transpose can be decomposed into a sum of $b$ matrices which are asymptotically free and identically distributed. We then consider the joint effect of different block decompositions and show that under some mild assumptions we also get asymptotic freeness.
We study Haar unitary random matrices with permuted entries. For a sequence of permutations $\left(σ_N\right)_N$, where $σ_N$ acts on $N\times N$ matrices we identify conditions under which the $\ast$--distribution of permuted Haar unitary matrices $U_N^{σ_N}$ is asymptotically circular and free from the unpermuted sequence $U_N$. We show that this convergence takes place in the almost sure sense. Moreover we show that our conditions on the sequence of permutations are generic in the sense that are almost surely satisfied by a sequence of random permutations.
We characterize the limiting fluctuations of traces of several independent Wigner matrices and deterministic matrices under mild conditions. A CLT holds but in general the families are not asymptotically free of second order and the limiting covariance depends on more information on the deterministic matrices than their limiting *-distribution.
We show that using the cyclic group the transpose of an R-cyclic matrix can be decomposed along diagonal parts into a sum of parts which are freely independent over diagonal scalar matrices. Moreover, if the R-cyclic matrix is self-adjoint then the off-diagonal parts are R-diagonal.
Using new combinatorial techniques, we significantly improve the previous results on asymptotic distributions and asymptotic free independence relations of partial transposes of Wishart random matrices. In particular, we give a necessary and sufficient condition for the asymptotic free independence of partial transposes of Wishart matrices with difference block sizes.
We present a combinatorial approach to the infinitesimal distribution of the Gaussian orthogonal ensemble (GOE). In particular we show how the infinitesimal moments are described by non-crossing partitions, but not of type B. We demonstrate the asymptotic infinitesimal freeness of independent complex Wishart matrices. With the combinatorial picture we can easily compute the infinitesimal cumulants of the GOE and demonstrate the lack of asymptotic infinitesimal freeness of independent GOE ensembles. I have added a new figure and some additional explanatory remarks. This update corrects a number of typographical errors; I am grateful to the readers who brought these to my attention.
We show that the partial transposes of complex Wishart random matrices are asymptotically free. We also investigate regimes where the number of blocks is fixed but the size of the blocks increases. This gives a example where the partial transpose produces freeness at the operator level. Finally we investigate the case of real Wishart matrices.
In this paper, we derive the bi-free analogue of the Lévy-Hinčin formula for compactly supported planar probability measures which are infinitely divisible with respect to the additive bi-free convolution introduced by Voiculescu. We also provide examples of bi-free infinitely divisible distributions with their bi-free Lévy-Hinčin representations. Furthermore, we construct the bi-free Lévy processes and the additive bi-free convolution semigroups generated by compactly supported planar probability measures.
We show that real second order freeness appears in the study of Haar unitary and unitarily invariant random matrices when transposes are also considered. In particular we obtain the unexpected result that a unitarily invariant random matrix will be asymptotically free from its transpose.